Papers

A family of heterotic compactifications carries more structure than a collection of solutions parametrised by moduli. Once the compactification data are fibred over moduli space, deformations become components of universal curvatures. This note reviews that organisation and explains how it incorporates the $\alpha'^2$ supersymmetry corrections.

In this paper, we construct several explicit examples of singular sets of Hausdorff dimension $(n-2)$ in $\mathbb{R}^n$ on free boundaries for an elliptic system modeling long range segregation. The system has been previously studied by Caffarelli, Patrizi and Quitalo in \cite{CL2} for the regularity of the free boundary in dimension two, and by the author and Torres in \cite{ChPaTo26_2} for the partial regularity in higher dimensions. However, the dimension of the singular set is unknown, and no concrete examples of singular set are known in the literature due to the nonlocal nature of the elliptic system. In this paper, we overcome this difficulty by rigidity and finite group action. As a byproduct of our result, we see that singular points can exist for the model in any dimensions. We also show that our method can be applied to the study of the singular set in the adjacent model. Finally, we also discuss some related open problems for future studies.

Motivated by the recent work of Lipton on the Möbius energy of helix curves, we extend the study to the $L^p$ asymptotics of the meromorphic family \[ M_\rho(t) = \frac{\rho^2+1}{\rho^2 t^2 + 4 \sin^2(t/2)} - \frac{1}{t^2}. \] The helix has infinite Möbius energy, but the arclength-rescaled energy density is finite. As $\rho \to 0$ the helix coils infinitely tight. Using contour integration and a careful Laurent expansion near the poles, we establish $I_p(\rho) := \left(\int_{-\infty}^\infty M_\rho(t)^p \, dt\right)^{1/p} \sim C_p \, \rho^{-(2-1/p)} $ for integer $p > 1$, extended to real $p > 1$, where $C_p$ is an explicit constant involving $\zeta(2p-1)$. The result gives the precise $L^p$ blowup rate of the Möbius energy density as the pitch $\rho \to 0$. The borderline case $p=1$ yields a logarithmic correction $I_1(\rho) \sim \log(1/\rho)/\rho$, recovering Lipton's main theorem. We derive a quantitative coiling barrier. Numerical verification confirms the scaling exponent to high precision.

Let $X$ be a Gorenstein minimal $3$-fold of general type whose canonical map is generically finite. We prove that if $p_g(X)> 243$, then the degree of the canonical map is at most $72$. Moreover, equality holds only if the general fibre $F$ of the Albanese morphism of $X$ is a smooth minimal surface of general type satisfying $p_g(F)=3,q(F)=0$ and $K_F^2=36$, and the canonical map of $F$ has degree $36$. This result improves the lower bound on $p_g(X)$ previously obtained by Jin-Xing Cai~\cite{Cai08}. As a consequence, we show that if the canonical degree is bigger than $64$, then the general fibre of the Albanese morphism of $X$ is a surface with irregularity zero.

In this paper, we propose a new view, which leads to almost sure well-posedness in $H^{s}(\mathbb{R}), s\geq 0$, for studying stochastic KdV equations. Different from \cite{de1999white} or \cite{kdvmuti}, by introducing a solution space inspired by \cite{guo2009global}, we prove the local well-posedness result only under natural $H^s(\mathbb{R}), s\geq 0$ conditions parallel to deterministic KdV equations. Furthermore, just basing on the $L_x^2$ conservation law of KdV equations, we extend the solution to a global one. The well-posedness frame obtained in this paper not only reduces several restrictions of the noise kernel, but may also have crucial values when one deals with dynamical problems of stochastic KdV equations.

In large scale X ray Computed Tomography (CT) inverse problems, the forward and back projectors are often generated using different discretizations. This discrepancy leads to unmatched pairs of projections, resulting in inconsistent normal equations. Consequently, employing the Conjugate Gradient method does not produce a useful solution. For matched operator pairs, the Golub Kahan bidiagonalization (GKB) method provides an efficient solution strategy. It works by projecting the original large-scale problem onto a lower-dimensional subspace, enabling the solution to be computed via a singular value decomposition of a sparse lower bidiagonal matrix. To address unmatched-pair problems in CT, we propose the AB and BA GKB algorithms as preconditioned forms of the GKB. These methods are straightforward to implement and allow for parameter tuning. We provide a discussion on the theoretical computational costs of our proposed algorithms in terms of floating point operations and compare with existing methods. While many Krylov methods tend to amplify noise in solutions, leading to semiconvergence, our proposed algorithms demonstrate greater resilience against this effect. We validate the effectiveness of our approach through numerical examples across various CT problems, showcasing its ability to deliver more stable solutions.

We propose and analyze a second-order consistent-splitting scheme, based on the generalized scalar auxiliary variable (GSAV) approach, for the two-dimensional perturbed Boussinesq system. The system is obtained by subtracting a stable, linearly stratified hydrostatic equilibrium from the standard Boussinesq system. The time discretization extends the consistent-splitting generalized BDF2 framework of Huang and Shen [17] for the Navier-Stokes equations, treating the nonlinear convection and advection together with the linear buoyancy and stratification couplings explicitly, so that each time step reduces to a small number of decoupled linear systems. We prove an unconditional weak stability theorem for the GSAV scheme and derive optimal second-order error estimates for the velocity, pressure, and temperature. A careful tracing reveals that the error constant depends on the inverse viscosity and inverse thermal diffusivity through a quadruply-nested exponential, so the scheme is not robust as either tends to zero. Numerical experiments confirm the second-order convergence and reproduce the expected internal-wave dynamics and exponential relaxation toward hydrostatic balance in a long-time stratified-flow simulation.

Fluid antenna systems (FAS) have emerged as a promising technology for next-generation wireless communications, offering inherent reconfigurability and spatial adaptability. A distinctive and practically consequential property of fluid antenna arrays (FAAs) is their geometric diversity: by dynamically activating different subsets of spatially distributed ports across a dense discrete grid, a FAA can reconfigure its effective aperture geometry on demand, thereby unlocking unprecedented spatial degrees of freedom for radiation pattern synthesis. Exploiting such geometric flexibility, this paper investigates peak sidelobe level (PSLL) minimization in sparse planar FAAs through enhanced heuristic optimization. Specifically, an improved genetic algorithm (IGA) is proposed to determine the optimal port activation pattern that minimizes the PSLL under strict sparsity constraints. The proposed IGA incorporates tournament selection, adaptive operator probabilities, a hybrid crossover scheme, multi-point mutation, and an elite-pool preservation strategy to improve both convergence speed and solution quality. Simulation results demonstrate that the IGA significantly outperforms the canonical GA (CGA) in convergence behavior and final PSLL performance, achieving a 4.45 dB reduction in sidelobe levels while maintaining a comparable mainlobe width.

We study a population model in which individuals carry one of two traits and evolve under mutation, selection, and density-dependent regulation. A deterministic large-population limit yields a nonlinear system coupling logistic growth with mutation-selection dynamics. We identify threshold conditions governing extinction, persistence, and long-term trait composition. In particular, mutation induces an effective mortality rate that determines whether the population can be sustained. When inheritance dominates mutation, a second threshold emerges: population establishment depends on initial trait composition as well as overall growth rates. Although extinction ultimately occurs, the system typically exhibits long-lived quasi-equilibrium behaviour. A diffusion approximation provides a tractable description of this, and reveals a transition in the sign of trait correlations. The model thus illustrates how mutation, selection, and resource limitation jointly shape both ecological persistence and evolutionary outcomes.

A partial Latin square of order $n$ is called $\epsilon$-dense if each row and each column contains at most $\epsilon n$ filled cells, and each symbol occurs at most $\epsilon n$ times. A partial Latin square is said to be completable if its empty cells can be filled to obtain a Latin square. Daykin and Häggkvist conjectured that every $\frac{1}{4}$-dense partial Latin square is completable. In this paper, we show that for all sufficiently large integers $n$, every $\frac{2}{25}$-dense partial Latin square of order $n$ is completable. The proof is obtained by establishing that there exists an $\eta > 0$ such that every triangle-divisible balanced tripartite graph on $3n$ vertices with partite minimum degree at least $(\frac{23}{25}-\eta)n$ admits a fractional triangle decomposition.

Each year, disruptions in the air transportation network strand millions of passengers and cost airlines billions in revenue. Airline networks prioritize operational and cost efficiency through hub-and-spoke structures that maximize revenue; however, these hubs also act as critical choke points during disruptions. Previous studies have focused on reactionary measures in response to air transportation network disruptions, whereas this work proposes a proactive strategy to improve resilience by reconfiguring the network's topology. Specifically, we consider airside-to-airside bus lines as a low-cost, frequent alternative to short, regional flights, offering service that can circumvent air traffic-related delays. We develop a network construction model that augments the existing air transportation network with these bus lines. The augmented networks are analyzed through an agent-based simulation, where increased resilience is measured in terms of decreased average hourly passenger delays under both nominal and disrupted conditions. Our results demonstrate that converting 10 regional routes from air service to airside-to-airside bus service, for a baseline scenario that is constrained by a $10 million investment budget, can reduce passenger delays by an average of 8% on disrupted days and 6% on nominal days. Furthermore, through a sensitivity analysis, we show that while augmenting the system using these buses decreases operational costs compared to the historic air-only network, continuously expanding bus parameters (i.e., range and investment budget) yields diminishing returns in delay mitigation. Finally, we discuss real-world precedents alongside regulatory and political hurdles to implementation. The proposed framework offers airlines, airports, and regulators a decision-support tool for integrating multimodal strategies into future disruption management policies.

In this paper, we investigate the application of fluid antenna systems (FAS) for active user detection (AUD) in unsourced random access (URA). A channel reconstruction method based on a one-dimensional convolutional neural network (1D-CNN) is proposed to effectively learn the nonlinear mapping from partial channel observations to the full channel vector. Furthermore, the reconstructed channel information is exploited to improve AUD performance via port selection. Simulation results demonstrate that the proposed 1D-CNN channel reconstructor significantly outperforms traditional methods under varying pilot lengths, achieving superior normalized mean squared error (NMSE) performance. Additionally, the reconstructed channel substantially reduces the AUD error rate compared with conventional approaches relying on traditional antenna configurations.

Analysing costs is crucial for optimising the operational efficiency and resource allocation in systems evolving under uncertainty. In this paper, we study the distribution of costs associated with the evolution of level-dependent quasi-birth-and-death (LD-QBD) processes, which are useful in modelling many multi-server systems. We derive analytical expressions for the Laplace-Stieltjes transforms (LSTs) of the distribution of total costs accumulated during the times the LD-QBD processes spend in a specified set of levels. We present algorithms for the numerical evaluation of these LSTs. We also give memory efficient versions of the algorithms and discuss their algorithmic complexity. To assess the robustness of the distribution of costs with respect to model parameters, we develop algorithms for the sensitivity analysis of the corresponding LSTs. To illustrate the application potential of our results, we construct LD-QBD example models for a finite capacity multi-server queueing systems with admissions policies including redirection, preemptive transfer, and guard-channel threshold. The analysis is based on a large dataset obtained from a tertiary referral hospital in Australia. We compute the long-run performance measures, the distribution of time until some number of beds become available following congestion, and the distribution of the associated costs. We present valuable insights into how the system behaves under the various policies. We also perform the sensitivity analysis of the distribution of costs with respect to model parameters.

We derive explicit formulas for the HOMFLY polynomials of the torus links $T(3,n)$ using braid groups and the skein relation. We first treat the case of $T(2,n)$ and then derive a five-term linear recurrence for an auxiliary sequence associated with $T(3,n)$. By solving this recurrence using a generating function, we obtain an explicit formula for the HOMFLY polynomial $P(T(3,n);y,z)$ of $T(3,n)$. The corresponding formula for $T(-3,n)$ is subsequently obtained from the mirror-image formula for the HOMFLY polynomial. As an application, we show that the HOMFLY polynomial distinguishes the links $T(3,n)$ within this family and distinguishes $T(3,n)$ from its mirror image for $n\geq 2$.

We propose a variational approach to principal spectral values of non-selfadjoint operator pencils $\mathcal L u=\lambda\mathcal G u$, where the weight operator $\mathcal G$ may be singular. The aim is to obtain Rayleigh-type minimax formulas for selected real spectral levels in settings where the standard selfadjoint variational theory is unavailable and positivity-based methods may not apply directly. The construction is based on the extended two-variable Rayleigh quotient \[ \mathcal R(u,v) = \frac{\langle \mathcal L u,v\rangle} {(\mathcal G u,v)_H},\] defined on admissible cone pairs. It leads to dual sup-inf and inf-sup principal levels, cone quasi-eigenvalues, and corresponding trapping and saddle-point principles. The resulting minimax formulas characterize selected real cone levels of non-selfadjoint operator pencils and identify them with principal spectral values whenever positive right-left eigenpairs exist, including cases with non-invertible operators and singular weights. We prove that these formulas are stable under finite-dimensional approximation. Thus the classical idea of approximating spectral data by finite-dimensional variational problems acquires an analogue for non-selfadjoint operator pencils in an ordered cone setting. The method also yields a posteriori spectral certificates, one-sided perturbation bounds, and approximation estimates. Elliptic examples illustrate both the scope of the method and the sharpness of the estimates.

The moduli space of meromorphic Higgs bundles admits a Poisson structure due to the independent work of Bottacin and Markman. In this paper, we revisit the symplectic leaves of this Poisson structure for the tame case. We study the partial compactification of the restricted Hitchin map on the symplectic leaves to an algebraically completely integrable system. In particular, we show that such a partial compactification is realized by the moduli spaces of $\vec{\xi}$-parabolic Higgs bundles. These same moduli spaces also provide a symplectic resolution of the normalization of the closure of the corresponding symplectic leaves. Finally, we discuss connectedness results for the corresponding Betti moduli spaces under the tame non-abelian Hodge correspondence.

This paper addresses the inverse problem of simultaneously recovering the fractional order $\alpha \in (0,1)\cup (1,2)$ and the time-dependent source factor $p(t)$ in the Cauchy problem for an evolution equation with a general self-adjoint operator $A$ in a Hilbert space $X$. The overdetermination condition is given by the scalar product $( u(t), \psi)_X$ for $0 < t < T$, where $\psi \in D(A)$ is an arbitrary fixed element. Uniqueness of the fractional order $\alpha$ is established independently of the specific form of the elliptic operator $A$ and the source function $p(t)$. Furthermore, uniqueness of the factor $p(t)$ is proved not only under the trivial overdetermination $( u(t), \psi)_X = 0$ for all $t \in (0,T)$, but also when the function $t \mapsto ( u(t), \psi)_X$ possesses sufficient smoothness. The proof relies on a decomposition of the solution near $t=0$ into a least smooth component and a smoother remainder.

We construct infinitely many embedded minimal hypersurfaces of pairwise distinct irreducible topological types in the unit $4$-sphere $\mathbb{S}^4$, which provides a new answer to a problem of Hsiang. These examples are topologically principal $S^1$-bundles and Seifert fibered manifolds over closed orientable surfaces. In particular, for any closed orientable surface $\Sigma_{2k-1}$ of odd genus $n=2k-1$, we show that $S^1\times \Sigma_{2k-1}$ admits a minimal embedding into $\mathbb{S}^4$. The construction is based on the equivariant min-max theory and the suspended (weighted) Hopf action on $\mathbb{S}^4$.

We prove small-data global stability of the Minkowski solution to Einstein's equations in a centre-normalised outgoing null-geodesic gauge. Our scheme involves first using the $r^p$-estimates of Dafermos-Rodnianski to control certain components of the Weyl tensor which satisfy a decoupled tensorial wave equation. Having established this control, all remaining geometric quantities are controlled by transport equations, taking initial conditions at a regular central axis. This method establishes global stability for initial data which decay only weakly to flat space and can establish additional asymptotic control when the data are assumed to have more structure.

We study Witt-vector affine Springer fibers for tame equi-valued conjugacy classes in tamely ramified groups. Following the approach of Goresky-Kottwitz-MacPherson in the equal characteristic setting, we show that they admit pavings by iterations of perfections of affine space bundles whose base are smooth Hessenberg varieties. Along the way we revisit a version of the Chevalley restriction theorem for the dual of Lie algebras.

This is an update of an expository article on the geometrization of orbital integrals of spherical Hecke functions on reductive groups over non-archimedean local fields, appeared in Proceedings of ICCM 2019. Compared to the published version, we add a last section on an example in SL3 case.

We study quantum speedups of derivative pricing for stochastic partial differential equation (SPDE) models through their backward doubly stochastic differential equation (BDSDE) representations. We develop conditional and nested quantum-accelerated multilevel Monte Carlo (QA-MLMC) methods for estimating the resulting conditional and nested expectations, improving the sampling complexity of classical Monte Carlo methods from $\widetilde{O}(\epsilon^{-2})$ to $\widetilde{O}(\epsilon^{-1})$ within additive error $\epsilon$. We apply the framework to derivative pricing and sensitivity analysis, providing quantum-accelerated estimators for prices as well as first-order and second-order Greeks, likelihood-ratio and Malliavin-weight representations for Greeks, and Heston-type stochastic-volatility models. To enable efficient multilevel coupling, we construct a family of Forward–Backward Taylor discretization schemes for the stochastic integrals arising in the BDSDE representations and establish global strong-error order one convergence for pricing and Greek estimators. Numerical experiments showcase our schemes for first-order and second-order Greeks can reach the required orders for the full quadratic quantum speedups.

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\mathcal A\subseteq 2^{[n]}$ is $r$-wise intersecting if $A_1\cap\cdots\cap A_r\neq\emptyset$ for all $A_1,\ldots,A_r\in\mathcal A$. In 2022, Tokushige proved that if $p_2<\frac{r-1}{r}$, then every $r$-wise intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf p}(\mathcal{A})\le p_1$, with equality only for stars centred at coordinates of maximum probability. He conjectured that the hypothesis $p_2<\frac{r-1}{r}$ can be replaced by $p_{r+1}<\frac{r-1}{r}$. In this paper, we prove this conjecture in full. The key novelty is the introduction of a new random partition method, which reduces the problem to at most $r$ coordinates and solves it exactly, thereby fully covering all cases with multiple supercritical coordinates.

We define log conifold transitions for Fano threefold pairs of index two and study their deformation theory. Relying on the recent solution to the relative Clemens conjectures in this setting, we construct rational curves with normal bundle $\OO(-1)\oplus \OO(-1)$ by blowing up anchored points on the boundary divisor. Contracting these curves yields a singular space with ordinary double points. We prove that local smoothings of the nodes can be lifted to global first-order deformations, and that the global deformation theory of both the log resolution space and the singular log pair is unconditionally unobstructed. Crucially, the geometry of the boundary del Pezzo surface guarantees this unobstructedness. Furthermore, unlike the classical Calabi-Yau case, the underlying Fano geometry forces the vanishing of global topological balancing conditions, allowing local first-order smoothings of the nodes to be lifted independently. As applications, we construct new non-Kähler threefolds via smoothings, we analyze the effective geometry of the smoothed threefolds by determining their Picard groups and proving the persistence of free curves. Finally, we study the Hodge theory of these non-Kähler threefolds.

Motivated by the increasing efforts to understand the covariant structure of physical models associated with General Relativity using different kinds of geometric frameworks, in this article we analyze the four-dimensional Palatini-Cartan model for gravity, which is a well-known generalization of General Relativity, from the perspective of various geometric-covariant formalisms for classical field theory. At the Lagrangian level, we do not only recover the correct field equations of the theory, which are equivalent to the torsion-free condition and the Einstein equations, but we also study the gauge symmetries of the model in order to construct the Lagrangian momentum map associated with the action of the gauge symmetry group on the configuration space of the system and, consequently, its corresponding Noether currents. Within the multisymplectic approach, we analyze the action of the gauge symmetry group on the multi-momenta phase space of the model, and we also introduce the induced momentum map that allows us to recover the admissible Cauchy data of the system. Further, we also apply the algorithm to treat singular systems within the polysymplectic framework, in which, in order to obtain the correct field equations of the model, we introduce a non-trivial Dirac-Poisson bracket characterized by the generalized Moore-Penrose inverse of the matrix induced by the second class constraints of the system. Finally, using the multisymplectic framework as a starting point, we perform the space plus time decomposition of the system to recover the instantaneous Lagrangian and the extended Hamiltonian of the theory, as well as the gauge structure that characterize the Palatini-Cartan model for gravity within the instantaneous Dirac-Hamiltonian formalism.

A factor of a graph is essentially a specific type spanning subgraph. In recent years, the spectral extremal problem of characterizing the existence of graph factors via eigenvalues has been widely studied. This paper focuses on fractional $ID$-$[a, b]$-factor-critical graphs, which are a natural generalization of fractional $[a,b]$-factors. Let $r \ge 1$ be an integer. A graph $G$ is fractional $ID$-$[a, b]$-factor-critical if for every independent set $I$ of $G$ with $|I| = r$, $G - I$ has a fractional $[a, b]$-factor. In 2026, Jia, Fan and Liu posed the spectral version conjecture for a graph to be fractional $ID$-$[a, b]$-factor-critical [Linear Algebra Appl. 732 (2026) 1-17]. In this paper, we first prove the conjecture holds for connected graphs when $b\ge 2r+2$. Furthermore, for minimum degree $\delta(G)\ge a+r$, we present spectral radius and size conditions that ensure a graph is fractional $ID$-$[a, b]$-factor-critical, which improve the results of Jia, Fan and Liu.

Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data. Existing algorithms for computing TT decompositions can be categorized into two main types: conventional batch-based approaches and recursive online methods. In the context of streaming data, batch methods typically achieve higher reconstruction accuracy but often suffer from memory exhaustion, while online methods provide greater computational efficiency. In this work, we introduce Online TT-ALS (Alternating Least Squares), an algorithm that sequentially enforces orthogonality constraints. This approach allows for efficient and exact updates of the core tensor while maintaining high reconstruction accuracy. Theoretically, we prove that enforcing these orthogonal gauge constraints guarantees monotonic decrease of the local objective function and temporal smoothness. Computationally, our deterministic single-sweep update reduces the rank dependence from quadratic to linear, achieving an overall complexity of $\mathcal{O}(I^{n-1} r)$. Experimental results demonstrate that the proposed method outperforms existing online techniques not only in terms of mathematical approximation accuracy but also in human perception-based video quality metrics. Furthermore, compared to recent deep learning-based paradigms, our algebraic approach achieves speedups of several orders of magnitude. Consequently, our method exhibits high computational efficiency and is suitable for low-latency real-time processing applications.

Wormlike micellar fluids exhibit complex rheological behavior driven by the continuous breakage and recombination of self-assembled micellar networks. Existing two-species models provide a coarse binary representation of the micellar population, limiting their ability to resolve intermediate structural states and broad relaxation spectra. To address this limitation, we develop a three-species cascade breakage model consisting of gel-network, long chains, and short chains. By introducing an intermediate micellar state, the model links the rapid relaxation of short fragments to the slow recovery of the gel-network within a unified kinetic framework. This additional structural pathway gives rise to a three-mode viscoelastic response, improves the high-frequency description of the dynamic moduli, and produces a non-monotone constitutive curve that evolves into a stress plateau with coexisting shear bands in Couette flow. This cascade mechanism also governs the transient response, including stress overshoot, hysteresis, and multistep relaxation after shear cessation. Overall, the proposed three-species model provides a physically interpretable framework for worm-like micellar shear banding, capturing the connection between cascade microstructural evolution, broad relaxation dynamics, and macroscopic flow localization.

We study high-frequency inference for an ergodic regime-switching jump diffusion with locally stable infinite-variation jumps. The state and the finite-state regime are observed on the same grid. The mesh satisfies \(h_n\to0\), while the horizon satisfies \(T_n=nh_n\to\infty\). We estimate structural drift and diffusion parameters in the presence of unknown regime-specific Lévy measures. For a common activity index \(1<\beta<8/5\), we remove two leading jump biases from each regime's truncated variation. We then combine the diffusion estimator with estimated signed tail means in an adaptive drift contrast. We prove joint asymptotic normality at rates \(\sqrt{T_n}\) and \(\sqrt n\), together with feasible studentization. The growing horizon requires uniform control of the debiasing remainders and random denominators over increasing regime exposures. It also produces an explicit Lévy-measure term in the drift covariance. A fixed-grid median stabilizes the two-step correction without changing its first-order limit. We study finite-sample behavior by simulation and apply the method to 127.5 hours of cattle sensor data. In the application, jaw-activity labels are recorded by a device separate from the sensor that measures the state process.

We prove that logarithmic capacity convergence for phase-union spectra of quasi-periodic Schrödinger operators in the zero Lyapunov exponent regime is robust, requiring only continuity of the potential. Let $S^+(p/q)$ denote the union, over the phase, of the spectra at rational frequency $p/q$. We show that if the Lyapunov exponent vanishes on the spectrum $\Sigma(\alpha)$ at an irrational frequency $\alpha$, then for every sequence $p_n/q_n\to\alpha$, the logarithmic capacities $Cap(S^+(p_n/q_n))\longrightarrow Cap(\Sigma(\alpha)).$ We also prove convergence of the corresponding harmonic measures. As a consequence, the equilibrium measures of $S^+(p_n/q_n)$ converge in the weak$^*$ topology to the density of states measure of the quasi-periodic Schrödinger operator. We extend these results to multi-frequency Schrödinger operators and prove analogous convergence theorems, for logarithmic capacity, harmonic measure, and equilibrium measure, for ergodic Schrödinger operators in a general setting where the almost sure spectrum is approximated in the Hausdorff metric by union spectra of periodic operators. This abstract formulation applies, in particular, to uniformly almost periodic potentials along sequences of almost periods. We also provide counterexamples when the limiting frequency is rational.